Theorem pm5.6 951

Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)

Assertion

Ref	Expression
pm5.6	⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Proof of Theorem pm5.6

Step	Hyp	Ref	Expression
1		impexp 462	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
2		df-or 385	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒))
3	2	imbi2i 326	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
4	1, 3	bitr4i 267	1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386

This theorem is referenced by:  ssundif  4052  brdom3  9350  grothprim  9656  eliccelico  29539  elicoelioo  29540  ballotlemfc0  30554  ballotlemfcc  30555  elicc3  32311  ifpidg  37836  icccncfext  40100